57 research outputs found
"Term Partition" for Mathematical Induction
A key new concept, {\em term partition}, allows to design a new method for proving theorems whose proof usually requires mathematical induction. A term partition of a term is a well-defined splitting of into a pair of terms that describes the {\em language of normal forms of the ground instances of }. If \A {\em monomorphic} set of axioms (rules) and (a,b)t\Aany ground instance of can be ``divided'' into the normal forms (obtained by using \Athe corresponding ground instances of abt = s\Aartition of and a partition of is computed. If and , then is an inductive theorem of \A The method is conceptually different to the classical theorem proving approaches since it tries to directly mechanize the \omega$-rule. It allows to obtain elegant and natural proofs of a large number of conjectures (including non-linear ones) without additional lemmas and/or generalizations
Sound Lemma Generation for Proving Inductive Validity of Equations
In many automated methods for proving inductive theorems, finding a
suitable generalization of a conjecture is a key for the success of
proof attempts. On the other hand, an obtained generalized conjecture
may not be a theorem, and in this case hopeless proof attempts for the
incorrect conjecture are made, which is against the success and
efficiency of theorem proving. Urso and Kounalis (2004) proposed a
generalization method for proving inductive validity of equations,
called sound generalization, that avoids such an over-generalization.
Their method guarantees that if the original conjecture is an
inductive theorem then so is the obtained generalization. In this
paper, we revise and extend their method. We restore a condition on
one of the characteristic argument positions imposed in their previous
paper and show that otherwise there exists a counterexample to their
main theorem. We also relax a condition imposed in their framework
and add some flexibilities to some of other characteristic argument
positions so as to enlarge the scope of the technique
Recent Results on the Periodic Lorentz Gas
The Drude-Lorentz model for the motion of electrons in a solid is a classical
model in statistical mechanics, where electrons are represented as point
particles bouncing on a fixed system of obstacles (the atoms in the solid).
Under some appropriate scaling assumption -- known as the Boltzmann-Grad
scaling by analogy with the kinetic theory of rarefied gases -- this system can
be described in some limit by a linear Boltzmann equation, assuming that the
configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) vol. 185
(1969), 308]). The case of a periodic configuration of obstacles (like atoms in
a crystal) leads to a completely different limiting dynamics. These lecture
notes review several results on this problem obtained in the past decade as
joint work with J. Bourgain, E. Caglioti and B. Wennberg.Comment: 62 pages. Course at the conference "Topics in PDEs and applications
2008" held in Granada, April 7-11 2008; figure 13 and a misprint in Theorem
4.6 corrected in the new versio
Mod-two cohomology of symmetric groups as a Hopf ring
We compute the mod-2 cohomology of the collection of all symmetric groups as
a Hopf ring, where the second product is the transfer product of Strickland and
Turner. We first give examples of related Hopf rings from invariant theory and
representation theory. In addition to a Hopf ring presentation, we give
geometric cocycle representatives and explicitly determine the structure as an
algebra over the Steenrod algebra. All calculations are explicit, with an
additive basis which has a clean graphical representation. We also briefly
develop related Hopf ring structures on rings of symmetric invariants and end
with a generating set consisting of Stiefel-Whitney classes of regular
representations v2. Added new results on varieties which represent the
cocycles, a graphical representation of the additive basis, and on the Steenrod
algebra action. v3. Included a full treatment of invariant theoretic Hopf
rings, refined the definition of representing varieties, and corrected and
clarified references.Comment: 31 pages, 6 figure
The periodic decomposition problem
If a function can be represented as the sum of
periodic functions as with
(), then it also satisfies a corresponding -order difference
equation . The periodic
decomposition problem asks for the converse implication, which may hold or fail
depending on the context (on the system of periods, on the function class in
which the problem is considered, etc.). The problem has natural extensions and
ramifications in various directions, and is related to several other problems
in real analysis, Fourier and functional analysis. We give a survey about the
available methods and results, and present a number of intriguing open
problems
Scaling up classification rule induction through parallel processing
The fast increase in the size and number of databases demands data mining approaches that are scalable to large amounts of data. This has led to the exploration of parallel computing technologies in order to perform data mining tasks concurrently using several processors. Parallelization seems to be a natural and cost-effective way to scale up data mining technologies. One of the most important of these data mining technologies is the classification of newly recorded data. This paper surveys advances in parallelization in the field of classification rule induction
Strategic Issues, Problems and Challenges in Inductive Theorem Proving
Abstract(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP
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